GEOMETRY* 269
ous directions. Jacobi determined the geodetic lines of the general ellipsoid. With the aid of elliptic coordinates (the parameters of three surfaces of a system of confocal surfaces of the second order passing through the point to be determined) he succeeded in Integrating the partial differential equation so that the equation of the geodetic line appeared as a relation between two Abelian Integrals, The properties of the geodetic lines of the ellipsoid are derived with especial ease from the elegant formulae given by Liou-ville. By Lame* the theory of curvilinear co-ordinates, of which he had Investigated a special case In 1837, was developed In 1859 Into a theory for space in his Lemons sur la thlorie des coordonntes curvilignes.
The expression for the Gaussian measure of curvature as a function of curvilinear co-ordinates has given an impetus to the study of the so called differential invariants or differential parameters. These are certain functions of the partial derivatives of the coefficients in the expression for the square of the line-element which In the transformation of variables behave like the invariants of modern algebra. Here Sauce", Jacobi, C. Neumann, and Halphen laid the foundations, and a general theory has been developed by Beltrami.* This theory, as well as the contact-transformations of Lie, moves along the border line between geometry and the theory of differential equations, f
* Mem. di Bologna, VIII. t Loria.